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How to Boil a Nucleus

The explosion of a supernova, a dying star, is an event of exceptional violence; it is the only phenomenon thought to be powerful enough to fuse atomic nuclei into heavy elements from iron up to uranium. During the catastrophic collapse of stellar material in a supernova’s core, gaseous nuclear matter is believed to condense to the liquid phase. A giant nucleus, or neutron star, is formed in this nuclear phase transition. It is composed primarily of neutrons, and it possesses the gross properties of a drop of water. A neutron star is an extraordinarily dense object, having the mass of our sun but a radius of only about 10 kilometers; nevertheless it has much in common with more conventional nuclei. Samples of nuclear “liquid” are abundant on earth. They are found in the nuclei of all heavy elements, where protons and neutrons swim in close proximity, bound by the strong nuclear force.

Recent experiments have begun to unravel the fluid properties of hot atomic nuclei. In giant accelerators, particle collisions heat the nuclear liquid and convert it to gas. This process, as it turns out, is effectively a reversal of the dramatic events at the core of a supernova collapse. By divining the properties of an atom’s nucleus as it undergoes a transition from liquid to vapor, scientists have an opportunity to understand the conditions required for the opposite phase transitionwhen the gaseous matter of a star condenses into a neutron star.

In order to understand the formation of neutron stars and black holes (objects even denser than neutron stars, created when extremely massive stars go supernova), it is essential to know the conditions under which nuclear matter changes from the gas to the liquid phase. These conditions are expressed, both for nuclear matter and ordinary matter, in terms of an equation of state, which describes the phase behavior of a substance as a function of temperature, pressure and composition. The equation of state for ordinary water describes the pressure and temperature at which molecules of steam condense to liquid water. A familiar example of an equation of state is the equation PV = mRT, the simplified expression for describing the behavior of gases as a function of pressure, volume, moles of gas, the gas constant R and temperature. This law is a simplified equation of state for describing the behavior of gases. Unfortunately, although it is possible to measure phase changes in chemical systems, we know the properties of atomic nuclei only near their stable ground states and at normal density. The conditions leading to the formation of a neutron star in the core of a supernova are much more extreme and inaccessible to direct measurements of temperature, pressure and composition and the relations of these properties to nuclear condensation.

Since we cannot directly measure the nuclear-matter condensation that occurs inside a supernova, the question becomes, how can we investigate such phenomena in our terrestrial environment? The only accessible approach is to study the reverse process, the expansion and vaporization of heavy nuclei. Fortunately, heavy atomic nuclei approach a neutron star’s density and share many of its fluid properties. The primary difference, of course, is that nuclei are much smaller and contain roughly equal numbers of neutrons and protons, whereas neutron stars are dominated by neutrons.

Transforming the nuclear liquid into gas can be achieved by bombarding a large nucleus such as gold (Au, atomic number 79) with a beam of energetic nuclear projectiles. We then are able to observe the vaporization products using particle detectors that measure the charges, sizes, velocities and other characteristics of the fragments, or “ejectiles,” that issue from a bombarded nuclei. The experimentalist’s challenge is to learn as much as possible about the fragments, and thereby infer the environment that created the event. Most importantly, are the fragments produced by shattering of the nucleus, sequential evaporation or true boiling?

The energy of particles necessary to create a nuclear phase transition is of the order of 2 GeV, or nearly a million times the energy of a typical medical x ray The unit eV, or electron-volt, is the kinetic energy imparted to a single charge accelerated by an electrical potential difference of one volt. The shorthand for a million electron-volts is MeV; for a billion electron-volts, GeV. Because energy and mass are interchangeable (following Einstein’s equation E=mc^sup 2^), the masses of particles also are commonly expressed in units of MeV and GeV. In the experiments we shall describe here, energy from accelerated beams of 3He ions, protons, pi mesons (bound states of quarks, the constituents of protons and neutrons) and anti-protons is used to heat liquid nuclei to the extreme temperatures at which a nuclear liquid-gas phase transition may take place (about 100 billion degrees Kelvin).

Some of the most dramatic work in this area involves the collision of two energetic heavy nuclei, for example gold ions. Such collisions produce compressional heating as the nuclear components, protons and neutrons (generically called nucleons), interweave to create a region of overlap, where the density exceeds that of normal matter. This compression front serves as a heating mechanism that dissipates energy throughout the combined system. Such studies currently are being conducted at the National Superconducting Cyclotron Laboratory (NSCL) at Michigan State University; the Grand Accelerator d’Ions Lourds (GANIL) in Caen, France; and the Gesellschaft fur Schwerionen Physik (GSI) in Darmstadt, Germany. Nevertheless, such systems also entail complications in studying the aftermath of nuclear heating. These include the dynamic effects of compression-decompression and high nuclear rotation, which store energy in forms other than heat. At much higher energies, experiments of this sort create densities of nuclear matter approaching that within a black hole and allow the investigation of temperatures at which nucleons may undergo another type of phase transition so profound that their constituent quarks intermingle to form quark matter, the state of matter presumably present about a millionth of a second after the Big Bang.

Boiling a Nucleus

To understand the phase transition in nuclear matter and to isolate the thermal properties of the equation of state, other studies employ an alternative method of nuclear heating: bombarding nuclei with light-ion projectiles. We employ this approach in studies using the Indiana Silicon Sphere (ISiS) detector, which sorts out the fragments resulting from “head-on” collisions of a light ion with a stationary target.

Instead of creating violent compressional heating, light ions, such as individual protons, anti-protons or helium nuclei, heat an atomic nucleus much as a microwave oven heats food. In a microwave oven, radiant energy is absorbed by the water molecules in food, causing them to vibrate and enter an excited state. As the molecules return to their normal state, they release stored energy as heat. Similarly, the collision of an accelerated charged particle with a nucleus induces a cascade of collisions among the constituent protons and neutrons. Some of these collisions excite the nucleons to very short-lived “resonant” states, which store specific (quantized) amounts of energy. To achieve the first excited, or A (delta), state, a nucleon absorbs about 290 MeV; to reach a higher threshold and enter the N* resonant state, the nucleon stores in excess of 500 MeV. In this way some of the kinetic enery of an incoming projectile can be absorbed and temporarily stored inside the nucleus.

The key to boiling a nucleus is the effective redistribution of the stored energy. After the energy of the light-ion impact initiates a cascade of internal collisions, the energy stored by the resonance states of the nucleons is released when they return to their ground state by releasing pi mesons, also commonly called pions. Each pion has a mass-energy equivalent of about 140 MeV, or roughly 15 percent of the mass of a proton or neutron. After they are released within the core of a nucleus, these pions have a high probability of being absorbed by other nucleons, releasing their mass-energy to the nucleus. Ultimately, however, the energy is distributed among all the particles of the nucleus, potentially heating it to the point of dissociation.

In order to visualize the evolution of these violent nuclear events, a computer simulation of a collision between a 5 GeV proton and a gold (^sup197^Au) nucleus was created by theorists Pawel Danielewicz and Wolfgang Bauer at Michigan State University. In this simulation, the proton approaches the center of the gold nucleus about 3 x 10^sup-23^ seconds after encountering the surface of the nucleus. During this period there is a slight compression of nuclear matter as multiple nucleon-nucleon collisions dissipate the energy of the proton and create a soup of protons, neutrons, resonant nucleons and pions.

As the proton drives through the core of the nucleus and exits after about 6 x 10^sup -23^ seconds, the resonant states decay to the ground state, emitting pions that rapidly heat the surrounding particles. At the same time, some nucleons are ejected in the wake of the proton. After about 10 x 10^sup-23^ seconds, the geometry of the nucleus is significantly disrupted, and the nuclear liquid may be said to be diluted. The simulations suggest that the ejection of nucleons may create a cavity in the core of the nucleus, which is surrounded by a spherical shell of the nucleons that remain. Because this shell contains positively charged protons, their mutual charge repulsion, in combination with energetic heating that has “boiled” the nuclear fluid, begins to tear apart the nucleus. This environment is conducive to the formation of nuclear clusters, fragments of the original gold nucleus, whose protons and neutrons recondense into the nuclei of lighter elements. Based on theory, these are the stages of nuclear boiling predicted by the computer simulation.

Theory and modeling set the stage for experiments to test their accuracy. Although it is possible for theorists using computer modeling to trace in exquisite detail the stepwise evolution of nuclear events at every stage, experimentalists do not have that luxury. At best, they can only know the properties of the target nucleus and the projectile that precede the collision and then measure its final products long after the reaction is over, nanoseconds later when the fragments reach the detectors (1 nanosecond = 10^sup -9^ seconds). This situation is analogous to the task of a forensic scientist attempting to reconstruct the explosion of a bomb based on the examination of debris collected at the scene of the blast.

Because understanding the vaporization of nuclear fluid demands a rigorous inventory of nuclear fragments, a versatile particle-detection system is needed to sort and classify the nuclear debris from a high-energy collision. Just such a system is the ISiS, a device that surrounds the collision site with three layers of particle detectors. The first layer uses an old technology, a gas-ionization detector similar to a Geiger counter. The unique sensitivity of ISiS, however, is derived from its use of modern silicon chips in the second layer and scintillator/photodiode technology in the third. These layers combine to make 162 “telescopes” comprised of combinations of detectors. Detector arrays for studies of heavy-ion collisions have been constructed at several institutions, but the ISiS has been dedicated to the examination of the kind of thermally induced, “soft” nuclear explosions under discussion. The particle detectors in the ISiS array cover 75 percent of the surface of a complete sphere. Based on the quantity of positive charge measured in each fragment, which reveals the number of protons in the fragment, ISiS is able to identify a range of elements from single-proton hydrogen to calcium, whose nucleus contains 20 protons. Because a gold nucleus contains a total of 79 protons, the combined charges of all the reaction fragments should add up to 79. How those charges are distributed reveals whether the collision has produced a few heavy nuclei or many smaller fragments, such as hydrogen and helium. ISiS also measures the kinetic energy of the fragments and the angle at which they are ejected from the exploding nucleus.

Boiling or Shattering?

One of the first problems the experimentalist must confront in sifting through the millions of events recorded in the detector array is determining the properties of the violent events that give rise to a disintegrating nucleus. Since the goal is to identify a phase transition, one must first demonstrate that the detector has witnessed a true “boiling” phenomenon, rather than the shattering effect of a proton acting like a cue ball hitting a rack of billiard balls. Both mechanisms leave the final system in a highly disrupted state, but only a true boiling event can tell us something about the nuclear equation of state. Various tests help distinguish between the extremes of boiling and shattering. For example, are the fragments emitted randomly in all directions, a signature of phase transition, or, consistent with a shattering mechanism, are they preferentially emitted in the direction of the incident projectile?

Based on the analysis of collisions that produce large numbers of fragments (multifragmentation), it appears that these events result from a relatively slow-moving heavy residual nucleus that has survived the initial projectile impact. The observed events are consistent with boiling or a “soft explosion” of the nucleus. However, most events also contain a few energetic fragments that travel in the general direction of the projectile, and these must be separated from the events that resemble the boiling phenomenon we seek to study.

Once an experimenter is sure that there is boiling, rather than shattering, going on, crucial questions about the nature of the phase transition can be asked. One question is whether the nucleus expands as it is heated. The physical parameter that describes this behavior, compressibility, is a critical quantity in the equation of state of any fluid. Knowledge of the compressibility of nuclear matter is central to understanding how supernovae condense to form neutron stars and black holes. Fluids that expand easily are said to have a “soft” equation of state in comparison to “hard” equation-of-state fluids such as water. Water actually contracts slightly as it is heated from freezing to 4 degrees Celsius and expands only about 1 percent by the time it reaches boiling at 100 degrees C.

Nuclear-scattering studies have examined normal nuclei near their ground states, indicating a relatively “soft” equation of state. However, our knowledge of compressibility is much less certain for a nucleus that is heated to its boiling point near 100 billion degrees Kelvin. Such temperatures have been reached by bombarding heavy nuclei with GeV protons and helium ions at the Alternating Gradient Synchrotron at Brookhaven National Laboratory and the Saturne II accelerator in Saclay, France. If liquid nuclear matter expands significantly as it is heated, then the repulsive electric-charge (Coulomb) interaction among the protons would be weaker, since they are farther apart, and fragments would be ejected at lower velocities. The repulsive charges possessed by the nuclear clusters contribute strongly to the tearing apart of the nucleus once its temperature is high enough to vaporize the nuclear liquid. Also, a true liquid would cease to expand when it reaches the temperature at which vaporization begins. In other words, the temperature should remain constant as the liquid is converted to gas at the boiling point, which would no longer result in the expansion of the liquid. Is this what happens to a nucleus as well?

In fact, both of these phenomena have been observed during the heating of nuclei with light ions. Examining the velocity of fragments emitted at various temperatures (based on collision violence) reveals that Coulomb repulsive forces indeed are reduced as temperature increases. Based on these observations, it can be deduced that a nucleus expands to a diameter about 50 percent greater than normal, its density dropping to about one-third that of normal nuclear matter and reaching a constant value after about 500 to 700 MeV of energy is deposited in the nucleus. This is near the heat of vaporization of these residual nuclei. The leveling-off of expansion at a specific range of temperatures is consistent with the behavior expected from a liquid undergoing a change of phase.

Further evidence for expansion and cooling of hot nuclei comes from analysis of pairs of fragments emitted from the same event-which provide a clock for examining the time evolution of the disintegration process. When two nuclear clusters are emitted in opposite directions, their relative velocities are sensitive to the size and temperature of the droplet of hot liquid that produces them. Large velocities imply that a small, hot source is pushing the fragments away; small relative velocities indicate a larger, cooler source. Studies of this phenomenon have shown that for carbon and heavier fragments, the system undergoes substantial expansion and cooling prior to breakup. Some fraction of the lighter elements, from hydrogen to boron, appear to be coming from a hotter, denser source. This suggests that the emission process is timedependent, with energetic light fragments being emitted early on as the system expands and cools, eventually followed by clusterization and repulsion-driven breakup of the nucleus into fragments of all sizes. Such a scenario is consistent with the expanding emittingsource model proposed by W. A. Friedman of the University of Wisconsin.

The Speed of Vaporization

Is disassembly a slow, sequential process, like the evaporation of a glass of water over time? Or is it nearly instantaneous, as in boiling? An answer to this question can be obtained by correlating the velocities of fragments with similar charges that are emitted at small relative angles to one another. If two fragments of the same velocity are emitted in the same direction at significantly different times, or at the same time with much different velocities, there will be little mutual electric-charge repulsion. Hence their trajectories will be only slightly affected. On the other hand, if two fragments going in the same direction with similar velocities are emitted at about the same time, then their trajectories will diverge as a result of the mutual repulsion of their nuclear charges. The degree of divergence as a function of fragment velocity allows one to determine whether or not a nucleus breaks up rapidly or slowly. Such analyses from the ISiS measurements indicate a breakup time of 10 x 10^sup -23^ to 20 x 10^sup -23^ seconds. This is much shorter that the typical time scale for the sequential evaporation of fragments from a hot nucleus (about 500 x 10^sup -23^). Combined with the observation that outgoing fragments have low energies, the behavior suggest that the final breakup stage indeed resembles “boiling.”

Armed with the results of these various analyses, it is possible to reconstruct the complex sequence of events that leads to the multifragmentation of hot nuclear systems formed in collisions with GeV light-ion beams. Initially, the nucleus is heated by a soup of colliding nucleons, pions and resonant states, trapping large amounts of energy in the nuclear interior. Some fast particles cascade out of the nucleus, leaving it in a state of depleted density, perhaps forming temporary bubble-like structures. This hot system then expands, emitting nucleons as well as boron and lighter fragments, and cooling in the process. Finally, when a density about one-third that of normal nuclear density is reached, the residue rapidly disintegrates into multiple clusters of nuclear matter. This final stage strongly resembles a phase transition, although such a conclusion still demands further investigation.

Nonetheless, it is tempting to speculate on this possibility. If we accept that the “soft explosions” observed in our detectors are indeed the result of a liquidto-gas phase transition, then it should be possible to derive a “heating curve,” analogous to that for water and representing the amounts of heat required to raise a fluid to a particular temperature. From analysis of the energies and charges of the fragments in a collision event, it is possible to derive its heat content or, in nuclear terms, the excitation energy per nucleon of the source.

The temperature of the nucleus can be estimated by means of a “thermometer” that depends on examining the ratios of isotopes for fragments of a given element, that is, noting what proportions of the fragments have various different numbers of neutrons in their nuclei. The tendency of a nucleus to emit different ratios of two isotopes is theoretically predicted to depend on how many of its quantum states are occupied, which in turn depends on the temperature of the system. Although still not well confirmed, isotope ratios offer some promise as nuclear thermometers and a means to correlate heat energy and temperature in nuclear matter.

The upshot of these investigations is that a class of collision events has been observed that exhibits many characteristics expected for a phase transition in nuclear matter. Nuclear expansion to a radius nearly 50 percent greater than for normal nuclear matter now seems to be well established, implying a rather soft equation of state for hot nuclei. Still, many questions demand further scrutiny. Are these rapidly evolving events really statistical in nature-an essential requirement for a phase transition? How well can we measure the temperature of the disintegrating system? Can fragments emitted at different times in the evolution of the reaction be isolated with better detection techniques? Continued research over the next few years will be directed at clarifying these issues. At present, the major task facing experimentalists and theorists alike is to make a quantitative connection between the data and the nuclear compressibility. Only then will it be possible to link multifragmentation studies on earth to the evolution of supernovae explosions in space.